#17030: Knot Theory as a part of GSoC 2014.
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Reporter: amitjamadagni | Owner: amitjamadagni
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.2
Component: algebraic | Resolution:
topology | Merged in:
Keywords: | Reviewers: Miguel Marco, Karl-
Authors: Amit Jamadagni, | Dieter Crisman, Frédéric Chapoton,
Miguel Marco | Travis Scrimshaw
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/ticket/17030 | 0cce5891b429ea267c45bd89adacff6ebb12e453
Dependencies: | Stopgaps:
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Comment (by mmarco):
I propose to use the word "unlinked components" for the case you are
talking about: they bound a disconnected surface, or, equivalently, they
could be represented in a planar diagram where there are no crossings that
involve two different components.
I chosed the word _isolated_components to reffer to that situation: where
the planar representation that we are dealing with has no crossings that
involve different components.
I considered other names, like "unlinked components", but I didin't like
them because the concept I wanted to consider does not deppend on the
3-dimensional topology, but on the particular planar representation that
we are using. If two components are shown as "isolated" in a planar
diagram (which is the case that is troubling us here), they are clearly
unlinked. But the converse is not true: we could have a very complicated
diagram with crossings everywhere that actually represents 2 unlinked
knots.
The implementation of the alexander polynomial assumes that the input
doesn't have isolated components (that is why we need to treat that case
separatedly), but is able to give the right output if there are several
unlinked components that are not isolated. So i guess that the solution of
detecting that case and giving a 0 output is the right one.
_isolated_components and _braid_word_components should do pretty much the
same. The difference is that one computes it from the PD representation
whereas the other uses the braid one.
So, can we give a positive review or are there still pending issues?
--
Ticket URL: <http://trac.sagemath.org/ticket/17030#comment:175>
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